Physics 834: Problem Set #10

Here are some hints, suggestions, and comments on the assignment.
Remember to keep track of the amount of time you spend doing the
(entire) assignment and record this number on your problem solution.

Recent changes to this page:

27-Nov-2011 --- added more suggestions to problem 1

25-Nov-2011 --- Original version.

Quantum mechanical spherical well.

You don't need to re-derive the radial part of the Laplacian
(although it won't hurt :), but show how it can be transformed
with the given potential to be the equation for spherical Bessel
functions. What is the difference in the equation
between when r is less than R
and greater than R? How does this change the solution to
the equation?
Be sure to note the general solution in each region and
why one of the two terms in each must have zero coefficient.
(If the wave function is to be normalizable, what are the
conditions at the origin and as r goes to infinity?)

This is completely analogous to solving the square well
problem in one dimension. Use the matching conditions to
determine possible eigenvalues. But it won't be in the form
asked for: use the recursion relations (which you can find
in either text --- be careful that signs are different for the
two types of spherical Bessel functions) to show that you
get Equation (3).

The l=0 and l=1 representations of the spherical Bessel
functions, including the modified Bessel functions, can be
referenced directly in some cases in Mathematica, but in all cases
can be defined in terms of the ordinary Bessel functions.
I recommend doing this.
Look up "Bessel
function" in the "Documentation Center" under Help and use the
formulas given in Arfken or Lea.
Since α=10 is given, this is just a numerical problem,
so you can use FindRoot. But always plot the function
first, so you know what root you are trying to find (don't
assume there is only one bound state!).

Damped oscillator Green's function by division-of-region.

To apply the division-of-region method, we need to solve the
homogeneous equation in each region (t < t' and t' < t)
with the relevant boundary conditions and then match the solutions
at t = t'. This is supposed to be a physical Green's function,
so it is the causal response at time t to an impulse
force at time t'. Before the force is applied, what do
you expect the response to be? The response is damped and the
driving impulse is only given at t', so what do
you expect for the response at large times?
We'll find this same Green's function by a transform method
in class (following section C.3 in Lea, so look here),
so you should know where
you are heading!

Here we just plug in the force into the usual integral of
the Green's function (like the examples). You are encouraged
to do this with Mathematica, including making a plot with sample
values of the parameters. Don't forget to include Assumptions
and any HeavisideTheta functions!

Neumann Green's function for one-d Helmholtz equation.
(This was formerly given for the Poisson equation, which had problems.)
You should be able to follow the lecture 18 notes, just changing the
boundary conditions for each of the methods. Look at the
"Green's functions, Part I" notebook on the Mathematica examples
page for a guide to making the plot. An easy check of your two
Green's functions is just to evaluate them directly for some test
x and x' values and see if you get the same result.
Don't forget the constant term in the expansion method!

Green's function for the diffusion problem.

What are the boundary conditions at t=0 and as
t goes to infinity?

Plot your answer at various different times to see if
it behaves as expected (a spreading gaussian).

Dirichlet Green's function for Poisson's equation.
Section C.7 is relevant here. What orientations of the
hemispheres do the two choices of angles represent? Use
the easier one to do the third part.